General Maths HSC Syllabus

General Mathematics Stage 6 Syllabus

GeneralMathematics

Stage 6

Syllabus

1


Original published version update:September 1999 – Board Bulletin/Official Notices Vol 8 No 7 (BOS 54/99)April 2000 – Board Bulletin/Official Notices Vol 9 No 2 (BOS 13/00)

May 2011 – Assessment and Reporting information updated

© 2011 Copyright Board of Studies NSW for and on behalf of the Crown in right of the State of New South Wales.

This document has been produced by the Board of Studies NSW on behalf of the Australian Curriculum, Assessment and Certification Authorities in collaboration with the Board of Studies Victoria, Curriculum Council of Western Australia, Northern Territory Board of Studies, Senior Secondary Assessment Board of South Australia, and the Tasmanian Secondary Assessment Board. The Material is protected by Crown copyright.

All rights reserved. No part of the Material may be reproduced in Australia or in any other country by any process, electronic or otherwise, in any material form or transmitted to any other person or stored electronically in any form without the prior written permission of the Board of Studies NSW, except as permitted by the Copyright Act 1968. School students in NSW and teachers in schools in NSW may copy reasonable portions of the Material for the purposes of bona fide research or study. Teachers in schools in NSW may make multiple copies, where appropriate, of sections of the HSC papers for classroom use under the provisions of the school’s Copyright Agency Limited (CAL) licence.

When you access the Material you agree:• to use the Material for information purposes only;• to reproduce a single copy for personal bona fide study use only and not to reproduce any major extract or the entire Material without the prior permission of the Board of Studies NSW; • to acknowledge that the Material is provided by the Board of Studies NSW; • not to make any charge for providing the material or any part of the Material to another person or in any way make commercial use of the material without the prior written consent of the Board of Studies NSW and payment of the appropriate copyright fee; • to include this copyright notice in any copy made; • not to modify the Material or any part of the material without the express prior written permission of the Board of Studies NSW.

The Material may contain third party copyright materials such as photos, diagrams, quotations, cartoons and artworks. These materials are protected by Australian and international copyright laws and may not be reproduced or transmitted in any format without the copyright owner’s specific permission. Unauthorised reproduction, transmission or commercial use of such copyright materials may result in prosecution.

The Board of Studies has made all reasonable attempts to locate owners of third party copyright material and invites anyone from whom permission has not been sought to contact the Copyright Officer, ph (02) 9367 8289, fax (02) 9279 1482.

Material on p 5 is from Securing Their Future © NSW Governemnt 1997.

Published by Board of Studies NSW GPO Box 5300 Sydney 2001 Australia

Tel: (02) 9367 8111 Fax: (02) 9367 8484 www.boardofstudies.nsw.edu.au

20110510

2

http://www.boardofstudies.nsw.edu.au/


Contents

1 The Higher School Certificate Program of Study................................................4

2 Rationale for General Mathematics in the Stage 6 Curriculum..........................5

3 Continuum of Learning for Stage 6 General Mathematics Students..................6

4 Aim .....................................................................................................................7

5 Objectives...........................................................................................................7

6 Course Structure................................................................................................8

7 Objectives and Outcomes..................................................................................9

7.1 Table of Objectives and Outcomes..........................................................9

7.2 Key Competencies.................................................................................11

8 Preliminary Course Content.............................................................................14

8.1 Financial Mathematics............................................................................15

8.2 Data Analysis.........................................................................................23

8.3 Measurement.........................................................................................32

8.4 Probability...............................................................................................40

8.5 Algebraic Modelling................................................................................44

9 HSC Course Content........................................................................................49

9.1 Financial Mathematics............................................................................50

9.2 Data Analysis.........................................................................................57

9.3 Measurement.........................................................................................64

9.4 Probability...............................................................................................70

9.5 Algebraic Modelling................................................................................74

10 Course Requirements.......................................................................................79

11 Post-school Opportunities................................................................................80

12 Assessment and Reporting..............................................................................81

3


1 The Higher School Certificate Program of Study

The purpose of the Higher School Certificate program of study is to: provide a curriculum structure that encourages students to complete secondary

education; foster the intellectual, social and moral development of students, in particular

developing their:– knowledge, skills, understanding and attitudes in the fields of study they choose– capacity to manage their own learning– desire to continue learning in formal or informal settings beyond school– capacity to work together with others– respect for the cultural diversity of Australian society;

provide a flexible structure within which students can prepare for:– further education and training– employment– full and active participation as citizens;

provide formal assessment and certification of students’ achievements; provide a context within which schools also have the opportunity to foster students’

physical and spiritual development.

4


2 Rationale for General Mathematics in the Stage 6 Curriculum

Mathematics involves observation, representation, investigation and comparison of patterns and relationships in social and physical phenomena. It allows the creative solution of problems and is also a powerful, precise and concise means of communication. At an everyday level, it is concerned with practical applications in many branches of human activity. At a higher level, it involves abstraction and generalisation. As such, it has been integral to most of the scientific and technological advances made in Australia and elsewhere.

Effective participation in a changing society is enhanced by the development of mathematical competence in contextualised problem-solving. Experience in such problem-solving is gained by students by:

(i) gathering, interpreting and analysing mathematical information and(ii) applying mathematics to model situations.

The opportunities for creative thinking, communication and contextualised problem-solving provided by this course, assist students to find positive solutions for the broad range of problems encountered in life and work beyond school.

General Mathematics supports the other Stage 6 mathematics courses in catering for the wide variation in students’ mathematical competence at the conclusion of Year 10, and provides opportunities for continuing mathematical growth.

The purpose of General Mathematics is to provide an appropriate mathematical background for students who wish to enter occupations which require the use of basic mathematical and statistical techniques. The direction taken by the course, in focusing on mathematical skills and techniques that have direct application to everyday activity, contrasts with the more abstract approach taken by the other Stage 6 mathematics courses.

The study of General Mathematics provides students with valuable support in a range of concurrent Stage 6 subjects, in fostering development of mathematical skills and techniques that assist students who undertake associated research and projects. The course provides a strong foundation for vocational pathways, either in the workforce or in further vocational training studies, and for university courses in the areas of business, the humanities, nursing and paramedical sciences.

5


3 Continuum of Learning for Stage 6 General Mathematics Students

6

Stages 1–3

K–6 Mathematics

Stage 4

Mathematics

Stage 5

Mathematics courses that provide relevant experiences for General Mathematics Stage 6:

Intermediate Mathematics Standard Mathematics

Stage 6

General Mathematics Preliminary and HSC

Workplace University TAFE Other

(selected courses)

Exp

erie

nce

in p

rob

lem

-solvin

g a

nd

mo

de

lling

as a

resu

lt of in

volve

me

nt in

cou

rses in

the

M

ath

em

atics L

ea

rnin

g A

rea


4 Aim

General Mathematics is designed to promote the development of skills, knowledge and understanding in areas of mathematics that have direct application to the broad range of human activity. Students will learn to use a range of techniques and tools to develop solutions to a wide variety of problems related to their present and future needs and aspirations.

5 Objectives

Students will develop:

appreciation of the relevance of mathematics the ability to apply mathematical skills and techniques to interpret practical situations the ability to communicate mathematics in written and/or verbal form skills, knowledge and understanding in financial mathematics skills, knowledge and understanding in data analysis skills, knowledge and understanding in measurement skills, knowledge and understanding in probability skills, knowledge and understanding in algebraic modelling.

7


6 Course Structure

The following schematic view provides an overview of the arrangement and relationship between components of the Preliminary course and HSC course for General Mathematics Stage 6.

Preliminary Course HSC Course

Financial Mathematics

FM1: Earning money FM2: Investing money FM3: Taxation

Financial Mathematics

FM4: Credit and borrowing FM5: Annuities and loan

repayments FM6: Depreciation

Data Analysis

DA1: Statistics and society DA2: Data collection and sampling DA3: Displaying single data sets DA4: Summary statistics

Data Analysis

(A) DA5: Interpreting sets of data(B) DA6: The normal distribution(C) DA7: Correlation

Measurement

M1: Units of measurement M2: Applications of area and volume M3: Similarity of two-dimensional

figures M4: Right-angled triangles

Measurement

M5: Further applications of areaand volume

M6: Applications of trigonometry M7: Spherical geometry

Probability

PB1: The language of chance PB2: Relative frequency and

probability

Probability

P1.1 PB3: Multi-stage eventsP1.1 PB4: Applications of probability

Algebraic Modelling

AM1: Basic algebraic skills AM2: Modelling linear relationships

Algebraic Modelling

AM3: Algebraic skills andtechniques

AM4: Modelling linear and non-linear relationships

8


7 Objectives and Outcomes

7.1 Table of Objectives and Outcomes

Objectives Preliminary Outcomes HSC Outcomes

Students will develop: A student: A student:

appreciation of the relevance of mathematics

P1develops a positive attitude to mathematics and appreciates its capacity to provide enjoyment and recreation

H1appreciates the importance of mathematics in her/his own life and its usefulness in contributing to society

the ability to apply mathematical skills and techniques to interpret practical situations

P2applies mathematical knowledge and skills to solving problems within familiar contexts

H2integrates mathematical knowledge and skills from different content areas in exploring new situations

P3develops rules to represent patterns arising from numerical and other sources

H3develops and tests a general mathematical relationship from observed patterns

skills, knowledge and understanding in algebraic modelling

P4represents information in symbolic, graphical and tabular forms

H4analyses representations of data in order to make inferences, predictions and conclusions

P5represents the relationships between changing quantities in algebraic and graphical form

H5makes predictions about the behaviour of situations based on simple models

skills, knowledge and understanding in measurement

P6performs calculations in relation to two-dimensional and three-dimensional figures

H6analyses two-dimensional and three-dimensional models to solve practical and mathematical problems

P7determines the degree of accuracy of measurements and calculations

H7interprets the results of measurements and calculations and makes judgements about reasonableness

9


Objectives Preliminary Outcomes HSC Outcomes

Students will develop: A student: A student:

skills, knowledge and understanding in financial mathematics

P8models financial situations using appropriate tools

H8makes informed decisions about financial situations

skills, knowledge and understanding in data analysis

P9determines an appropriate form of organisation and representation of collected data

H9develops and carries out statistical processes to answer questions which she/he and others have posed

skills, knowledge and understanding in probability

P10performs simple calculations in relation to the likelihood of familiar events

H10solves problems involving uncertainty using basic principles of probability

the ability to communicate mathematics in written and/or verbal form

P11justifies his/her response to a given problem using appropriate mathematical terminology

H11uses mathematical argument and reasoning to evaluate conclusions drawn from other sources, communicating his/her position clearly to others

10


7.2 Key Competencies

General Mathematics provides a context within which to develop general competencies considered essential for the acquisition of effective, higher-order thinking skills necessary for further education, work and everyday life.

Key competencies are embedded in the General Mathematics Stage 6 Syllabus to enhance student learning. The key competencies of collecting, analysing and organising information and communicating ideas and information, reflect core processes of statistical inquiry and are explicit in the objectives and outcomes of the syllabus. The other key competencies are developed through the methodologies of the syllabus and through classroom pedagogy. Students work as individuals and as members of groups to engage with applications and modelling tasks, and through this, the key competencies planning and organising activities and working with others and in teams are developed. At all levels of the course, students are developing the key competency using mathematical ideas and techniques. Through the advice provided on the selection and use of appropriate technology, students can develop the key competency of using technology. Finally, students’ continual involvement with seeking solutions to problems, both large and small, contribute towards their development of the key competency solving problems.

11


Presentation of Content

The course content is presented in five areas of study, aspects of which are studied in each of the Preliminary and HSC courses. Within each area of study, the material is divided into cohesive units of work, each of which contributes to the students’ achievement of one or more of the course outcomes. It is intended that the prescribed skills, knowledge and understanding be developed through the study of appropriate tasks and applications that clearly demonstrate the need for such skills.

Units of work are presented in the following format:

Name of Unit

Focus statement: a brief summary of content and purpose of the unit.

Outcomes addressed

An indication of the specific course outcomes that will be informed through study of the unit.

Students learn and acquire the following skills, knowledge and understanding

An identification of the mandatory mathematical content of the unit.

Terminology introduced in this unit

An identification of words or phrases that may be new to students, and which may be used in relevant assessment tasks. Students are expected to understand these terms but will not be asked to define them.

Technology that may be used in support of this unit

The provision of advice about the nature and suggested use of technology that is appropriate to the unit.

Suggested applications and modelling tasks

The provision of examples of a range and style of applications that may be used to introduce and illustrate the mathematical content of the unit. These applications are not mandatory, and are presented here for the support of teachers. The selection provided in the syllabus is necessarily brief. Further examples and resources for modelling activities will be found in the support document.

12


The use of formulae in the teaching and assessment of the course

Any formulae that have been required in the Mathematics Stage 4 (Years 7–8) Syllabus are considered to be assumed knowledge. These include formulae for:

calculations involving the theorem of Pythagoras perimeter the circumference of a circle the area of a rectangle, triangle and circle.

Formulae required in the themes of the Stage 5 Standard course are considered to be assumed knowledge. These include formulae for the:

area of a parallelogram, trapezium and rhombus volume of a right prism.

Students are not required to learn other formulae that are introduced or referred to in this syllabus. A list of formulae will be provided with the HSC examination. This list is included with the sample HSC examination that accompanies this syllabus.

13


8 Preliminary Course Content

Hours shown are indicative only.

Financial Mathematics 24 hours FM1: Earning moneyFM2: Investing moneyFM3: Taxation

Data Analysis 32 hours

DA1: Statistics and societyDA2: Data collection and samplingDA3: Displaying single data setsDA4: Summary statistics

Measurement 32 hours M1: Units of measurementM2: Applications of area and volumeM3: Similarity of two-dimensional figuresM4: Right-angled triangles

Probability 16 hours

PB1: The language of chancePB2: Relative frequency and probability

Algebraic Modelling 16 hours

AM1: Basic algebraic skillsAM2: Modelling linear relationships

14


8.1 Financial Mathematics

FM1: Earning money

The principal focus of this unit is the range of ways in which individuals earn and manage their money.

Outcomes addressed

A student:

P1 develops a positive attitude to mathematics and appreciates its capacity to provide enjoyment and recreation

P2 applies mathematical knowledge and skills to solving problems within familiar contextsP7 determines the degree of accuracy of measurements and calculations P8 models financial situations using appropriate toolsP11 justifies his/her response to a given problem using appropriate mathematical

terminology.


calculation of monthly, fortnightly, weekly, daily and hourly payments from salary calculation of wages incorporating hourly rate, penalty rates such as overtime, special

allowances for, for example, wet work, confined spaces, toxic substances, heat, heights calculation of annual leave loading calculation of earnings based on commission, piecework, royalties calculation of income based on government allowances, such as youth allowance,

pensions determination of deductions such as union fees, superannuation contributions, health

fund instalments and tax instalments calculation and comparison of user costs associated with maintaining accounts with

financial institutions calculation of net pay following deductions creation and management of budgets reading information from household bills, including those for electricity, gas, telephone,

council rates and water rates.

15



annual leave annual leave loading deductions double time gross pay/gross wage overtime royalty time-and-a-half.


use of a prepared spreadsheet to calculate earnings use of a prepared spreadsheet to create a budget.

Suggested applications and modelling tasks (FM1)

Use a spreadsheet to calculate wages for a small company including the calculation of overtime.

Calculate earnings based on current Enterprise Agreements. These can be obtained from unions or the human resource department of an organisation.

Review a previously prepared budget to reallocate funds for a sudden contingency.

16


FM2: Investing money

The principal focus of this unit is to use formulae and tables to perform calculations related to the value of investments over a period of time.

Outcomes addressed

A student:

P2 applies mathematical knowledge and skills to solving problems within familiar contexts P3 develops rules to represent patterns arising from numerical and other sourcesP7 determines the degree of accuracy of measurements and calculationsP8 models financial situations using appropriate toolsP11 justifies his/her response to a given problem using appropriate mathematical

terminology.


calculation of simple interest using , where P = principal, r = percentage interest rate per period expressed as a decimal (eg if the rate is quoted as 8.2%, then r = 0.082), and n = number of periods

P1.1 for fixed values of P, using tables of values and hence drawing and describing graphs of I against n for differing values of rNote: these are linear graphs whose gradient is determined by the value of r(see AM2, AM4)

calculation of monthly, quarterly, six-monthly interest rates based on quoted rates per annum (pa)

use of formulae to calculate future value, compound interest and present value with pen and paper

, where A (amount) = final balance (future value), P (principal) = initial quantity (present value), n = number of compounding periods, r = interest rate per compounding periodNote: In the financial world, the compound interest formula quoted above is generally presented as , where FV = future value and PV = present value

17


for fixed values of P, using tables of values and hence drawing and describing graphs of A against n for differing values of rNote: these are examples of exponential growth (see AM3, AM4)

calculation of dividend paid on a share holding and the dividend yield, excluding franked dividends

extrapolating from the information shown on a prepared graph of share performance to suggest possible future movement

calculating future and present value of an investment from prepared tables(see table below)

calculation of the price of goods following inflation calculating the appreciated value of items such as stamp collections, memorabilia.

Compounded values of $1

Interest rate per periodPeriods 1% 5% 10% 15% 20%1 1.010 1.050 1.100 1.150 1.2002 1.020 1.103 1.210 1.323 1.4403 1.030 1.158 1.331 1.521 1.7284 1.041 1.216 1.461 1.750 2.0745 1.051 1.276 1.611 2.011 2.488 6 1.062 1.340 1.772 2.313 2.986

Eg: to calculate the future value of $3000 invested for 6 years compounded annually at 5%,

A = 3000 x 1.340 = $4020

18



compounding perioddividenddividend yieldfuture value present valueshares.


use of a spreadsheet’s chart tool to create graphs of functions based on the compound interest formula, eg compare the value of an investment of $1000 over 5 years at 10% pa compounded (a) annually(b) bi-annually(c) monthly.


If the interest rate is quoted as 6% pa, what amount needs to be invested in order for the investment to be worth $850 at year’s end?

Jan and Bob wish to save $10 000 for their granddaughter’s university expenses. They wish to have this amount available in 8 years’ time. Calculate the single sum to be invested at 5% pa compounded annually.

Determine the single sum to be deposited if $10 000 is required in 5 years’ time and terms of 3% pa (compounded quarterly) are available.

A principal of $1000 is to be invested for three years. Determine which is the better investment option: (i) 6% pa simple interest, (ii) 5.9% pa compounded annually or (iii) 5.85% pa compounded half-yearly.

Use a spreadsheet to create tables to determine the future and present values of $1. Students could access and analyse historical data on share prices. This data is

available from financial journals, newspapers and the Internet.

19


An investor purchases 1000 shares in a company at a price of $3.98, with a dividend yield of 5.5%. Brokerage costs are 1% of the purchase price. One year later, the shares sold at $4.80. Calculate the total earnings over the year, after costs.

Use published share data to create a table of a company’s share price over a period of time.

Use the charting facilities of a spreadsheet to illustrate performance of a company’s shares over a period of time.

Students could create and track a portfolio of shares and publish their results each month. The value of the portfolio could be compared to the changes in the All Ordinaries Index in the time period examined.

Students could participate in the ‘ASX Game’. This game is administered by the Australian Stock Exchange. Players nominally invest $50 000 in listed shares. This could occur in conjunction with other subjects, with the emphasis, in this instance, on mathematics and modelling activities. For details visit the Australian Stock Exchange website.

20


FM3: Taxation

The principal focus of this unit is the calculation of tax payable on income and goods and services.

Outcomes addressed

A student:

P2 applies mathematical knowledge and skills to solving problems within familiar contextsP5 represents the relationships between changing quantities in algebraic and graphical

formP7 determines the degree of accuracy of measurements and calculationsP8 models financial situations using appropriate toolsP11 justifies his/her response to a given problem, using appropriate mathematical

terminology.


calculation of the amount of allowable deductions from gross income calculation of taxable income calculation of Medicare levy (basic levy only — see Tax Pack for details) calculation of PAYE (Pay As You Earn) tax payable or refund owing, using current tax

scales given rates of tax from a range of countries, calculation of the Value Added Tax (VAT)

payable on a range of goods and services calculation of the goods and services tax (GST) payable on a range of goods and

services creating graphs to illustrate and describe different tax rates

Note: a graph of tax paid against taxable income is a piecewise linear function (see AM2).


goods and services tax (GST)group certificateincome taxMedicare levyPAYEtax deductiontaxable incomevalue added tax (VAT).

21



use of a prepared spreadsheet to calculate PAYE tax (see example below).


Calculate tax refund (or amount payable) based on a sample Group Certificate, taking into account gross income, deductions, taxable income, tax payable on taxable income, Medicare levy and tax already paid as per the Group Certificate.

Students could complete a tax return form (as included in the Tax Pack) using a typical PAYE employee’s earnings and deductions. The aim is to calculate the ‘refund from’ or ‘amount owed to’ the Australian Taxation Office (ATO).

22


8.2 Data Analysis

DA1: Statistics and society

The focus of this unit is the importance of statistical processes and inquiry in society. The material treated is essential background for the later units in Data Analysis, and may be integrated with the later units if desired.

Outcomes addressed

A student:


P9 determines an appropriate form of organisation and representation of collected dataP11 justifies his/her response to a given problem using appropriate mathematical

terminology.

Students learn about and acquire the following skills, knowledge and understanding

the importance of analysing data in planning and decision-making by governments and businesses

the process of statistical inquiry, including the following steps:– posing questions– collecting data– organising data– summarising and displaying data– analysing data and drawing conclusions– writing a report

the role of statistical methods in quality control in manufacturing industries issues of privacy and ethics in data collection and analysis organisations that collect and/or use statistics, including the Australian Bureau of

Statistics (ABS), the United Nations (UN), the World Health Organisation (WHO).


datainformationquality controlstatisticsstatistical inquiry.

23


Suggested applications and modelling tasks (DA1)

Explore some of the different forms of information collection and recording that have been and are being used — the tally stick, church records, census, sport and weather information, the Guinness Book of Records, computer networks, data banks, creditratings …

Investigate the role of statistics in shaping and describing aspects of life in our society (eg forecasting economic trends, forecasting weather, consumer opinion polls, public perception surveys, TV/radio surveys).

Investigate the contribution made to the field of statistics by people such as:– John Graunt – Carl Friedrich Gauss– Florence Nightingale– Sir Ronald Fisher– W. Edwards Deming.

24


DA2: Data collection and sampling

The principal focus of this unit is the planning and management of data collection. In some cases, complete data sets may be available, and in other cases a survey may be needed that involves the use of a sample. Although the emphasis is on quantitative data, students should be aware of processes related to categorical data.

Outcomes addressed

A student:


P9 determines an appropriate form of organisation and representation of collected data

P11 justifies his/her response to a given problem using appropriate mathematical terminology.


identification of the target population to be investigated

determining whether data for the whole population is available (eg the results of a round of a sporting competition), or whether sampling is necessary

recognising that the purpose of a sample is to provide an estimate for a particular population characteristic when the entire population cannot be accessed

classification of data as:– Quantitative, either discrete or continuous– Categorical

eg gender (male, female) is categorical; height (measured in cm) is quantitative, continuous; quality (poor, average, good, excellent) is categorical; school population (measured in individuals) is quantitative, discrete

distinguishing between the following sample types:– random– stratified– systematic

determination of which of the above sample types is appropriate for a given situation

relating sample selection to population characteristics, eg if 20% of the Australian population is aged under 20, your sample should include 20% of under 20s

generating random numbers with a table or a calculator to assist in establishing random samples

describing and using the ‘capture-recapture’ technique for estimating the size of populations, eg the number of fish in a lake

recognising the effect of sample size in estimating the nature of the population, eg using the number of boys and girls in a particular Year 11 class to estimate the gender ratio in Year 11 across NSW

25


use of the principles for effective questionnaire design, such as– simple language– unambiguous questions– respect for privacy– freedom from bias– consideration of number of choices, if given, eg an even number of choices may

force an opinion in particular circumstances, but on occasions it may be appropriate to allow a neutral choice.


bias categorical data census continuousdatabase discrete poll populationquestionnaire random sample sample size stratastratified sample systematic sample.

Suggested applications and modelling tasks (DA2) Prepare and present oral and written reports to describe collected data, including the

source of the data and the type of data. If a sample is used, describe how the sample was chosen and how the measurements were made.

Prepare questionnaires and discuss consistency of presentation and possible different interpretations of the questions.

Discuss when it is appropriate to use a sample rather than a census. Discuss and present conclusions drawn from published data. Discuss and decide on methods of random number generation. Determine the best method of sampling in a range of situations. Discuss the reasonableness of drawing conclusions about populations from particular

samples (eg whether it is appropriate to draw conclusions about the whole school from data based on a single class).

Discuss and make judgements about arguments in the media for which statistical information is presented.

Discuss the difference between a census and a sample, and illustrate with examples of where samples and censuses are used.

Consider different ways of presenting questions, eg open questions, yes/no questions, tick boxes, response scales versus continuum scales etc.

Discuss bias, the representativeness of the sample chosen and other issues that may affect the interpretation of the results.

Select a range of samples from a fixed population and record the characteristics of each sample.

The ‘capture-recapture’ technique can be simulated by using a large (but uncounted) number of toothpicks in a bowl to represent the population of fish in a particular waterway. Discuss why it may not be practical to catch all of the fish in order to count them. Take a small number (10 or 20) of the toothpicks and mark them with a coloured pen. Return them to the bowl and shake them around. This simulates the capture, tagging and release of a known number of fish. Now take a handful of the toothpicks (fish) from the bowl and establish the percentage of tagged fish in the handful. Since the total number of tagged fish is known, this percentage may be used to estimate the size of the total population.

26


DA3: Displaying single data sets

In this unit, students prepare a variety of data displays and consider the appropriateness of each for the stated purpose. The power of statistical displays both to inform and misinform should be emphasised.

Outcomes addressed

A student:P1 develops a positive attitude to mathematics and appreciates its capacity to provide

enjoyment and recreationP4 represents information in symbolic, graphical and tabular formsP7 determines the degree of accuracy of measurements and calculationsP9 determines an appropriate form of organisation and representation of collected dataP11 justifies his/her response to a given problem using appropriate mathematical

terminology.


creation of tally charts and frequency tables to organise ungrouped and grouped data creation of dot plots, sector graphs (pie charts), bar graphs, histograms and line graphs,

with attention being paid to the scale on each axis selection of a suitable scale for each axis of a graph noting the capacity of statistical displays for misrepresentation, particularly in the

selection of the scale used on the axes creation of a stem-and-leaf plot to illustrate a small data set drawing a radar chart to display data such as sales figures, temperature or rainfall

readings (see example at end of unit) division of data into deciles and quartiles determination of the range and interquartile range as measures of the spread of a data

set creation of frequency graphs and cumulative frequency graphs (ogives) determining the median and upper and lower quartiles of a data set from a cumulative

frequency polygon establishment of a five number summary for a data set (lower extreme, lower quartile,

median, upper quartile, upper extreme) development of a box-and-whisker plot from a five number summary linking types of data with appropriate displays, eg continuous quantitative data is best

represented by a histogram; categorical data is best represented with a bar graph or sector graph (pie chart)

describing the strengths and/or weaknesses of sector graphs, bar graphs, histograms, frequency polygons and radar charts, including suitability for data represented.


use of a spreadsheet or a graphing calculator to create frequency tables and statistical graphs

27



bar graph box-and-whisker plot column graph cumulative frequency decile dot plotfive number summary frequency table grouped data histogram interquartile range line graph ogive polygon quartile radar chart range relative frequencysector graph stem-and-leaf plot upper/lower extremeupper/lower quartile.

Suggested applications and modelling tasks (DA3) Prepare rough dot plots for initial exploration of data.

Suggest possible reasons for the occurrence of any unusual scores in a data set.

Having gathered data, use pen and paper, a scientific calculator, graphing calculator and/or spreadsheets, other appropriate software or databases to organise the data and draw graphs.

Decide on the most appropriate type(s) of display for a particular data set, choosing from a histogram, stem-and-leaf plot or other type of graph.

Change the type of display on the computer to see if a different impression is given of the shape of the data display.

Perform experiments and record and analyse data to determine whether or not performance of a simple task can be improved with practice.

Compare and contrast the suitability of different types of statistical displays, eg a radar chart can be used to illustrate seasonal change or hourly change of a quantity over one day; a line graph is useful to show trends in data over equal time intervals.

Students could collect examples of misleading statistical displays and prepare accurate versions of each. They could describe the inaccuracies, and their corrected version, in appropriate mathematical language.

28


DA4: Summary statistics

The principal focus of this unit is the calculation of summary statistics for single data sets and their use in interpretation.

Outcomes addressed

A student:

P2 applies mathematical knowledge and skills to solving problems within familiar contextsP4 represents information in symbolic, graphical and tabular formsP7 determines the degree of accuracy of measurements and calculationsP11 justifies his/her response to a given problem using appropriate mathematical

terminology.


calculation of the mean of small data sets, using the formulae

, where , represents the mean of the sample

determination of the mean for larger data sets of either grouped and ungrouped data using the statistical functions of a calculator

calculation of the means of a range of samples from a population informal description of standard deviation as a measure of the spread of data in relation

to the mean

determination of the population standard deviation using the button of a calculator and the sample standard deviation as an estimate of the population measure, using the

button

determination of the median and mode(s) of a data set, either from a list or from a frequency table

selection and use of the appropriate statistic (mean, median or mode) to describe features of a data set, eg median house prices, modal shirt size

comparison of the summary statistics of various samples from the same population.


meanmedianmodestandard deviationsummary statistic.

29



calculate summary statistics from inbuilt statistical functions of a graphing calculator use a spreadsheet to create frequency tables and calculate summary statistics


Investigate the use of the word ‘average’ in a range of social contexts and the use of different ‘averages’ to serve different purposes, and sometimes, to mislead.

Interpret and evaluate data from students’ own surveys and draw conclusions that can be justified.

Report orally and in writing on investigations, discussing what was investigated, how the investigation was planned and the data collected, the display and the analysis, together with the conclusion that could be drawn.

Write a letter to a newspaper summarising the results of an investigation, suggesting the implications, and explaining and justifying conclusions.

30


8.3 Measurement

M1: Units of measurement

In this unit, the principal focus is on metric units of measurement, and rates and ratios. The students learn about making judgements about measurement errors.

Outcomes addressed

A student:

P2 applies mathematical knowledge and skills to solving problems within familiar contextsP5 represents the relationships between changing quantities in algebraic and graphical

formP7 determines the degree of accuracy of measurements and calculations.


determination of appropriate units to use when measuring physical attributes conversion between commonly used units of measurement using standard prefixes

recognition that accuracy of physical measurement is limited to of the smallest unit

of which the measuring instrument is capable calculation of the percentage error in a measurement, eg if the measured height was

155 cm ± 0.5 cm (ie to the nearest cm),

the percentage error for this measurement is

determination of possible sources of error in measuring repeating and averaging measurements to reduce likelihood of error determination of the significant figures to be used in recording measurements, in

relation to the accuracy of the measuring instrument being used use of positive and negative powers of ten in expressing numbers in scientific notation calculation of rates eg pay rates, speeds, rates of flow conversion between units for rates, eg km/h to m/s calculation of concentrations expressed as weight/weight, weight/volume or

volume/volume Note: these calculations have particular applications to nursing and agriculture determination of overall change in a quantity following repeated percentage changes,

eg an increase of 20% followed by a decrease of 20% finding the ratio of two quantities in familiar contexts division of quantities in a given ratio use of unitary method to solve problems.

31



concentrationpercentage errorprefixratiosignificant figuresunitary method.

Suggested applications and modelling tasks (M1)

Students measure their heights and calculate the percentage error in their measurements.

Modify given recipes by varying quantities to provide for various numbers of people. Applications from nursing may be considered, eg a patient fed intravenously by means

of drops of fluids. The patient needs 3 L of fluid per day. One mL of fluid contains 15 drops. Find the rate at which the intravenous drip must run, expressing the answer in the form of number of drops fed to the patient each minute.

Calculate the quantity of each component for a fertilising operation, given the ratio of each component in the mixture.

Calculate and compare freight costs for a variety of modes of transport. Calculate the rates of application of chemicals used in agriculture, such as those for

pesticides and feed additives. Your vet has recommended a vitamin supplement, for your sick guinea pigs, at the rate

of 10 g per litre in their drinking water. From the chemist you can obtain the vitamin supplement in solution and the label states ‘300 g/300 mL’. How many mL of the solution should you add to the guinea pigs’ 500 mL water bottle?

32


M2: Applications of area and volume

The principal focus of this unit is the calculation and application of area and volume in the solution of problems.

Outcomes addressed

A student:

P2 applies mathematical knowledge and skills to solving problems within familiar contextsP6 performs calculations in relation to two-dimensional and three-dimensional figuresP7 determines the degree of accuracy of measurements and calculations.


calculation of the area of triangles and quadrilaterals (review only) using a field diagram to calculate the area of irregularly shaped blocks of land classifying polyhedra into prisms (named with respect to their constant cross-section),

pyramids or other construction of nets of solids and matching nets to solids sketching 3D solids using isometric paper and vanishing points using appropriate formulae in calculating surface area of right prisms, square and

rectangular pyramids using appropriate formulae in calculating volume of right prisms, cylinders, pyramids,

cones, spheres application of the relationship between units of capacity and units of volume.

33



use of drawing tools to create 2D and 3D drawings.


Use appropriate software applications for drawing accurate figures and investigating geometrical properties.

Survey an irregular area using offsets and use this information to calculate its approximate area.

Discuss and report on possible sources of error, eg experimental, instrumental and constant error in the above activity.

Estimate the painted surface area of a classroom. Plan and prepare a budget for the redecoration of a favourite room at home.

Investigate the dimensions that maximise the area for a given shape and perimeter, such as in the design of playpens and stock paddocks.

34


M3: Similarity of two-dimensional figures

The principal focus of this unit is to apply similarity properties to problems in everyday life.

Outcomes addressed

A student:

P2 applies mathematical knowledge and skills to solving problems within familiar contextsP6 performs calculations in relation to two-dimensional and three-dimensional figuresP7 determines the degree of accuracy of measurements and calculations.


establishment of properties of similar figures recognition of similarity in everyday life finding scale factors of similar figures recognising that similar figures related by a scale factor of 1 are said to be congruent use of the relevant enlargement or reduction factor to calculate actual dimensions development of scale drawings of objects and images use of scale factor to solve problems involving similar figures transferring measurements between floor plans and elevations obtaining measurements from plans of buildings and rooms calculation of lengths and areas from a floor plan interpretation of commonly used symbols on house plans.


congruentelevationenlargementreductionscale factorsimilar.

35


Suggested applications and modelling tasks (M3) Investigate the scale factor of enlargements obtained using an overhead projector. Is

there a relationship between the distance of the projector from the screen and the scale factor of the resulting projection?

By measuring the shadow thrown by a metre rule, students use similarity and shadow lengths to find the height of tall objects, eg tree, flag pole etc.

Students accurately construct a scaled floor plan of the classroom. Students collect floor plans of new houses (from newspapers, new home exhibition

centres…) and discuss features of the houses that appeal to them or features that they would like to change. They could compare the cost of the houses with their floor areas and determine cost per square metre. From this they could determine the most expensive/economical house to build. Students could complete this in groups and present their findings to the class.

Find ceiling heights from plans. Use house plans to cost carpeting, tiling, painting rooms etc. Use a grid over a free-form diagram to draw an enlargement or reduction. Undertake an investigation relating to scale drawings and area, eg design a space-

efficient car park to hold a certain number of cars. Students could construct scale models of cars and the car park to ensure that there is enough space for vehicles to reverse park, move safely about the car park etc.

Use the properties of its diagonals to determine whether or not a room is rectangular. Students build similar rectangular prisms and cubes, finding the volume of each. They

could then tabulate results and establish patterns. Students could investigate what happens to costs associated with area, surface area

and volume if lengths are doubled, tripled…? For investigation: If all the dimensions on earth doubled overnight, how would you know

when you got up the next morning?

36


M4: Right-angled triangles

In this unit, students learn to solve practical mathematical problems involving right-angled triangles.

Outcomes addressed

A student:

P2 applies mathematical knowledge and skills to solving problems within familiar contextsP3 develops rules to represent patterns arising from numerical and other sourcesP6 performs calculations in relation to two-dimensional and three-dimensional figuresP7 determines the degree of accuracy of measurements and calculationsP11 justifies his/her response to a given problem using appropriate mathematical

terminology.


use of Pythagoras’ theorem to find an unknown side in a right-angled triangle application of Pythagoras’ theorem to:

– determine whether or not a triangle is right-angled– solve problems based on single right-angled triangles– calculate perimeters of irregularly shaped blocks of land

defining sine, cosine and tangent ratios use of trigonometric ratios to find the length of an unknown side in a right-angled

triangle use of trigonometric ratios to find the size of an unknown angle in a right-angled triangle

using a calculator to approximate the angle to the nearest minute solution of problems involving angles of elevation and depression, given the appropriate

diagram determining whether an answer seems reasonable by using a diagram drawn roughly in

proportion.


adjacentangle of elevationhypotenusesinetrigonometry.

angle of depression cosine oppositetangent

37



Find possible lengths of two sides of a right-angled triangle, given the length of the hypotenuse.

Investigate the trigonometric ratios for angles of, say, 30°, 45°, 60° in a number of similar right-angled triangles.

In groups, students use a clinometer to find the heights of school buildings etc. Different groups compare and discuss their answers. They could use angles of elevation or depression or use the fact that tan 45° = 1.

Construct and use a ‘height measurer’ to measure the heights of trees. Using a metre rule leaning against a wall along with a blackboard protractor, students

measure, as a decimal fraction of a metre:(a) the height the ruler reaches up the wall;(b) the distance of the foot of the ruler from the wall, for a range of different angles.

Plan and carry out an orienteering event at the school.

38


8.4 Probability

PB1: The language of chance

In this unit, students learn to use the language of probability, count outcomes and describe the sample space of an event.

Outcomes addressed

A student:


P3 develops rules to represent patterns arising from numerical and other sourcesP10 performs simple calculations in relation to the likelihood of familiar eventsP11 justifies his/her response to a given problem using appropriate mathematical

terminology.


ordering everyday events from the very unlikely to the almost certain using a list or table to identify the sample space (set of all possible outcomes) of a

simple experiment or game performing experiments and determining whether or not the outcomes are equally likely determining the number of outcomes for a multi-stage event by multiplying the number

of choices at each stage, eg the total number of ways to place 3 different letters in 3 envelopes is 3 x 2 x 1

using systematic lists to verify total number of outcomes for simple multi-stage events. Note: factorial notation is not required.


equally likely eventfrequencymulti-stage eventoutcomessample space.

39


Suggested applications and modelling tasks (PB1)

Practical experiments could involve coin tossing, dice rolling, picking cards from a pack. Comment critically on statements involving probability, such as: ‘Since it either rains or

is fine, the probability of a fine day is 50-50’. What is a ‘one in 300-year flood’? Investigate expressions used in other disciplines and

in everyday life to describe likely or unlikely events, eg ‘once in a blue moon’. Statements involving the language of probability could be collected from various media

(newspapers, magazines, radio, television, the World Wide Web) and discussed. Investigate the number of different meals that can be chosen from a menu. Determine the number of combinations of raised dots that are possible in the Braille

system for reading and writing. Investigate whether or not they are all used. Repeat for Morse Code.

Investigate whether we will ever run out of postcodes, car numberplates or phone numbers.

40


PB2: Relative frequency and probability

The main focus of this unit is to compare relative frequency and calculated probability.

Outcomes addressed

A student:

P2 applies mathematical knowledge and skills to solving problems within familiar contextsP4 represents information in symbolic, graphical and tabular formsP10 performs simple calculations in relation to the likelihood of familiar eventsP11 justifies his/her response to a given problem using appropriate mathematical

terminology.


estimating the relative frequencies of events from recorded data performing simple experiments to obtain relative frequencies from recorded results using relative frequencies to obtain approximate probabilities using the following definition of the probability of an event where outcomes are

equally likely:

calculating probabilities in terms of the fractional, decimal, or percentage chance

demonstrating the range of possible probabilities, , through examination of a variety of results

comparing calculated probabilities with experimental results illustrating the results of experiments through statistical graphs and displays (see DA3) defining and using the relationship between complementary events


complementary eventspercentage chance probabilityrelative frequency.

41



Data could be generated from simple experiments, and also be obtained from other sources where experiments are not possible, eg weather and sporting statistics from the newspaper, other data from Australian Bureau of Statistics (ABS) Yearbooks, websites.

Statements involving the language of probability could be collected from various types of media (newspapers, magazines, radio, television, Internet) and discussed.

If it is appropriate for the school population, plan a school raffle for fundraising. Experiments could be carried out in which the probability is not intuitively obvious, eg

P(drawing pin landing point up). Examine the birth notices on a particular day in a daily paper. Record the number of

boys and the number of girls. On this basis, estimate the probability that a child born is (a) male or (b) female. Compare these results with those published by ABS.

42


8.5 Algebraic Modelling

AM1: Basic algebraic skills

The main focus of this unit is to provide a foundation in basic algebraic skills required for this area of study and to use these skills to solve real and abstract problems.

Outcomes addressed

A student:

P2 applies mathematical knowledge and skills to solving problems within familiar contextsP3 develops rules to represent patterns arising from numerical and other sourcesP7 determines the degree of accuracy of measurements and calculations.


identification and generalisation of simple linear number patterns adding and subtracting like terms evaluation of the subject of a formula through substitution of numerical values, using a

wide variety of formulae such as

C =59

F −32( ) , v =u+at, S =DT

, V =43πr3 , A=P 1+ r( )n , B=

mh2

, D=mA150 ,

D=yA

y+12( ) , D=kA70

solution of linear equations involving up to 3 steps (fractions with numerical denominators only)

expansion and simplification of expressions such as

3p p −2( ) , 2x2 7−x( ) + x x−1( )

multiplication of algebraic terms such as 3M2 ×5M , 9y

4×5y

division of single terms (linear, quadratic and cubic) such as 9L3

3L2

solving equations following substitution and evaluation,

eg find a, given that v = 10, u = 5, s = 8 and v2 =u2 + 2as.


cubiclinearpowerquadratic.

43


Suggested applications and modelling tasks (AM1)

‘Practical’ formulae for evaluation, eg C =59

F −32( ) , v =u+at, S =DT

,

v2 =u2 + 2as, V =43πr3 , h2 =a2 +b2 .

Teachers may decide to introduce some of the formulae from the Financial Mathematics section here to emphasise how new variable names may be used, for example, FV =PV 1 + r( )n . Note that FV is a single number, the Future Value of an investment, and PV is a single number, the Present Value.

Algebra can be used to describe numerical patterns in calendars: take any 2 x 2 square, the sum of the diagonals is the same. To describe this, call the top left hand number x, then the number below it is x + 7 , the number to its right is x +1 , the last number in the square is x +1( ) + 7 . The result for the diagonals can be described as x + x+1( ) + 7 = x+1( ) + x+ 7( ) .Other patterns can be found in larger squares and represented algebraically.

Linear number patterns can be found in matchstick activities such as ‘line of squares’. Trade examples: A formula for calculating the bend allowance in sheet metal is

B=2π R+T1

⎛ ⎝ ⎜ ⎞

⎠ ⎟×

A360 , where B is the bend allowance or arc length, T is the thickness in

mm, A is the number of degrees in the angle of the bend, and R is the radius of curvature in mm.

Fried’s rule gives the medicine dosage for an infant in comparison with the

adult dosage: D=mA150

where D = infant dosage, m = age of infant in months

and A = adult dosage. Two similar formulae are Young’s rule: D=yA

y+12( )

where y = age of child in years, and Clark’s rule: D=kA70

where k = mass of child in

kilograms. Compare the results of each rule for an 18-month-old child of mass 8.5 kg, if the adult dosage of the particular medication is 10 mL.

For adults, the Body-Mass Index B=mh2 where m = mass in kilograms and h = height in

metres. The medically accepted healthy range for B is 21≤B≤25 . Calculate the value of B for Carlo, who is 1.8 m tall and weighs 90 kg. If Carlo were to consult you as his doctor, what positive suggestions might you make to him about lifestyle?

44


AM2: Modelling linear relationships

Outcomes addressed

A student:

P3 develops rules to represent patterns arising from numerical and other sourcesP4 represents information in symbolic, graphical and tabular formsP5 represents the relationships between changing quantities in algebraic and graphical

form.


sketching graphical representations of quantities that vary over a period of time or in relation to each otherNote: Students should develop an understanding of a function as input processing output. It is not intended that students learn a formal definition of a function.

identifying independent and dependent variables in practical contexts graphing of linear functions derived from everyday situations (eg cost of an excursion =

fixed cost + cost per student x number of students) by plotting ordered pairs from tables of values

calculating the gradients of such graphs with ruler and pencil establishing a meaning for the gradient in the given context establishing a meaning for the intercept on the vertical axis in the given context sketching graphs of linear functions expressed in the form y=mx+b development of a linear graph of the form y=ax from a description of a situation in

which one quantity varies in a direct linear fashion with another, given one ordered pair using the above graph to establish the value of a (the gradient) and to solve problems

related to the given variation context interpreting linear functions as models of physical phenomena using stepwise and piecewise linear functions to model situations encountered in daily

life, eg parking charges, taxi fares, tax payments, mobile phone bills recognising the limitations of such models, eg a person’s height as a function of age

may be approximated by a straight line for a limited number of years, but not over a complete lifetime

using graphs to make conversions from one measurement to another eg $AUD to Euros

interpreting the graphical solution of simultaneous linear equations drawn from practical situations

drawing a line of best fit on a graphed set of ordered pairs with a ruler and pencil.

45



decreasing function dependent variable direct linear variation extrapolatefunction gradientincreasing function independent variable interpolate piecewise linear function* proportional to rate of change relation step function* y-intercept.

* in practical contexts only


use of a graphing facility to: graph linear functions determine graphically the intercepts using ‘zooming’ techniques and built-in commands find points of intersection graphically inspect ordered pairs on a graph by ‘tracing’.


Functions and associated mathematical concepts to be introduced via everyday examples, such as:

– distance travelled as a function of time elapsed

– cost of postage as a function of mass (a step function)

– amount of fine for speeding as a function of amount by which speed limit is exceeded (a step function)

– a person’s height as a function of age (not strictly linear but can be approximated as a linear function)

– cost of a mobile phone plan as a function of time and/or as a function of the number of calls

– income tax rate as a function of taxable income (a piecewise function).

Practical examples of linear functions include:

– printing costs which involve an initial setup cost and a dollar rate per item (or hundreds of items…) printed

– cost of taxi fare which may be approximated by hiring charge + (dollar rate per kilometre) x (number of kilometres travelled). Hiring charge is the vertical intercept and the gradient is the dollar rate per kilometre

– catering costs which may involve a base amount for a set number of people plus a rate for extra guests

– drawing conversion graphs (usually lines through the origin).

Compare different mobile phone plans (graphically) to determine the most economical plan for a person’s likely pattern of use.

46


Blank Page

47


9 HSC Course Content

Hours shown are indicative only.

Financial Mathematics 30 hours

FM4: Credit and borrowingFM5: Annuities and loan repaymentsFM6: Depreciation

Data Analysis 26 hours

DA5: Interpreting sets of dataDA6: The normal distributionDA7: Correlation

Measurement 24 hours

M5: Further applications of area and volume M6: Applications of trigonometry M7: Spherical geometry

Probability 16 hours

PB3: Multi-stage eventsPB4: Applications of probability

Algebraic Modelling 24 hours

AM3: Algebraic skills and techniquesAM4: Modelling linear and non-linear relationships

48


9.1 Financial Mathematics

FM4: Credit and borrowing

This unit of work focuses on the mathematics involved in borrowing money, the different types of loans available and credit cards.

Outcomes addressed

A student:

H1 appreciates the importance of mathematics in her/his own life and its usefulness in contributing to society

H2 integrates mathematical knowledge and skills from different content areas in exploring new situations

H5 makes predictions about the behaviour of situations based on simple modelsH8 makes informed decisions about financial situationsH11 uses mathematical argument and reasoning to evaluate conclusions drawn from other

sources, communicating his/her position clearly to others.


calculation of principal, interest and repayments for flat-rate loans calculation of values in a table of home loan repayments (see below)

Home Loan Table

Amount = $50 000 This table assumes the same number of days in each month,ie Interest = Rate/12 x Principal

Annual Interest Rate = 10%

Monthly Repayment(R) =

$600

N Principal (P) Interest (I) P + I P + I - R

1 $50 000 $416.67 $50 416.67 $49 816.67

2 $49 816.67 $415.14 $50 231.81 $49 631.81

3 $49 631.81 $413.60 $50 045.40 $49 445.40

4 $49 445.40 $412.05 $49 857.45 $49 257.45

5 $49 257.45 $410.48 $49 667.93 $49 067.93

comparison of different options for borrowing money in relation to total repayments, fees, interest rates and flexibility

calculation of credit-card payments, incorporating fees, charges, rates and interest-free periods

use of published tables from financial institutions to determine monthly repayments on a reducing balance loan.

49


Terminology introduced in this unitflat rate loanreducing balance loanterm of loan.


use of a prepared spreadsheet to simulate loans naming of cell ranges to make the construction of formulae easier to understand and

implement use a graphing calculator to calculate, graph and analyse loan repayments.


Construct and use a simple loan spreadsheet initially by using paper, pen and calculator and then by using spreadsheet software. Consider car loans, travel loans, loans for capital items, home loans and others.

Use a spreadsheet to create a loan-repayment table for amount borrowed versus interest rate.

Calculate the effective interest rate (E) charged on a loan in relation to the stated rate

(r), using where E = effective interest rate per period, expressed as a

decimal. Students could use a home loan spreadsheet to vary the amount borrowed, the interest

rate and the repayment amount. Determine the answers to suitable ‘what-if’ questions, eg ‘How much can the interest rate rise before the amount of the repayment has to be increased?’ ‘What would the effect of this be on the term of the loan?’

Use a prepared graph of ‘amount outstanding’ against ‘repayment periods’ to determine when a particular loan will be half-paid.

50


FM5: Annuities and loan repayments

The principal focus of this unit is the nature and mathematics of annuities; the processes by which they accrue and the ways of maximising their value as an investment. Emphasis should be placed on using formulae and tables.

Outcomes addressed

A student:



H5 makes predictions about the behaviour of situations based on simple modelsH8 makes informed decisions about financial situations H11 uses mathematical argument and reasoning to evaluate conclusions drawn from other



recognition that an annuity is a form of investment involving periodical, equal contributions to an account, with interest compounding at the conclusion of each period

calculation of the future value (A) of an annuity (or the contribution per period), using

where M = contribution per period, paid at the end of the period.

Note: the future value of an annuity is the total value of the investment at the conclusion of the last period for payment.

For example, I am planning to take the trip of a lifetime in ten years’ time and estimate that the amount of money I will need at that time is $30 000. I am advised to contribute $2500 each year into an account that pays 4% pa, compounded annually. Will I have enough money in ten years time to make my dream come true? By how much will I fall short of or overshoot my goal?

51


calculation of the present value (N) of an annuity (or the contribution per period), using

Note: the present value of an annuity is the single sum of money which, if invested today at the rate of compound interest which applies to the annuity, would produce the same financial result over the same period of time.

For example, which would give the better financial result at the end of 20 years — a lump sum of $100 000 invested today at 12% pa compounded annually, or a monthly payment of $1 000, commencing today, with interest of 12% pa compounded monthly?

using tables to solve problems involving annuities use the present value formula for annuities to calculate loan instalments, and hence the

total amount paid over the term of a loan investigate various processes for repayment of loans calculate the fees and charges which apply to different options for borrowing money in

order to make a purchase.


annuityfuture value of an annuitypresent value of an annuity.


use the financial functions of a spreadsheet or graphing calculator to solve problems involving annuities

use a graphing facility or graphing calculator to develop graphs to illustrate the growth of an annuity

use spreadsheet software or a graphing calculator to create, model and analyse graphs and tables of loans

use a graphing calculator to model financial situations.


At the end of 12 years, a netball club needs to replace goal posts and other club equipment. It is estimated the replacement cost will be $10 000. How much will the club need to deposit each year at 5% interest?

Jo deposits $100 each month in her account for 10 years. How much does she accrue if the bank pays 7% interest compounded on a half-yearly basis?

Superannuation problems.

52


Use the table of future values of $1 of an ordinary annuity (see table below) to calculate the value of an ordinary annuity of $300 per month which is invested at the rate of 2% per month for 5 months.

Use a spreadsheet to create the table. Calculate the number of periods needed to repay a specified loan, using ‘guess and

refine’ methods. Investigate a ‘low-start’ home loan where the repayments are initially low because the

principal/interest has been allowed to accrue or the interest rate is capped for an initial period of time. Assess and describe the negative aspects of these types of loans.

Future values of $1

Interest RatePeriod 1% 2% 3% 4% 5%

1 1.0000 1.0000 1.0000 1.0000 1.00002 2.0100 2.0200 2.0300 2.0400 2.05003 3.0301 3.0604 3.0909 3.1216 3.15254 4.0604 4.1216 4.1836 4.2465 4.31015 5.1010 5.2040 5.3091 5.4163 5.52566 6.1520 6.3081 6.4684 6.6330 6.80197 7.2135 7.4343 7.6625 7.8983 8.14208 8.2857 8.5830 8.8923 9.2142 9.5491

53


Use a prepared spreadsheet to compare various home loans (see sample spreadsheet). The spreadsheet should allow the user to vary the amount borrowed, the rate, the repayment and the term of the loan.

Draw a graph showing how the amount still owing is related to the term of the loan. Use the present value formula to calculate the monthly repayment on a home loan of

$100 000 at 12% pa compounded monthly for 20 years. Hence calculate the total repayments and total interest over the term of the loan.

54


FM6: Depreciation

The focus of this unit is to investigate situations involving the depreciation of an asset over time.

Outcomes addressed

A student:



H5 makes predictions about the behaviour of situations based on simple modelsH8 makes informed decisions about financial situations H11 uses mathematical argument and reasoning to evaluate conclusions drawn from other



modelling depreciation by using appropriate graphs, tables and functions using formulae for depreciation:

a) the straight line method, where S = salvage (current) value of asset, D = amount of

depreciation apportioned per period, V0 = purchase price of the asset, andn = total number of periods

b) the declining balance method, where S is the salvage value after n periods, V0 is the purchase

price of the asset and r is the percentage interest rate per period, expressed asa decimal

preparing tables of values and hence developing graphs of against n for different values of rNote: these are examples of exponential decay (see AM3, AM4)

comparing the results obtained through each method using the above formulae to create and compare depreciation tables calculating tax deductions based on depreciation of assets.


assetdeclining balance methoddepreciationsalvage valuestraight line method.

55



use a spreadsheet to model the depreciation of an asset over time use technology to graph data for a depreciating asset and apply an equation of best fit

to the data. Use this equation to predict the value of the asset in the future.


Students could collect price data for a depreciating asset over time and use technology to determine the depreciation rate. For example, students could collect twelve prices (one per month) of a particular model and make of car over an eight year period. Students could determine the median price for each year and then create a scatter plot of the median price versus the year. The current year’s price could be taken as the car’s initial value. By changing the value of r in the equation, students could determine the graph that best fits the scatter plot and so determine r, the depreciation rate. This could be achieved on a graphing calculator by letting X represent r and Y represent V.

Zheng purchases a computer on March 1 for $5000. Calculate the amount that can be used as a tax deduction in this financial year if the declining balance method is used and the depreciation rate is 30% pa.

Read and interpret published tables of item costs, eg the NRMA publishes these tables for motor vehicles.

Students may compare graphs of the depreciation of an item. The value of the item would be read from the vertical axis and time read from the horizontal axis.

56


9.2 Data Analysis

DA5: Interpreting sets of data

The principal focus of this unit is the use of data displays, measures of location and measures of spread to summarise and interpret one or more sets of data.

Outcomes addressed

A student:



H4 analyses representations of data in order to make inferences, predictions and conclusions

H5 makes predictions about the behaviour of situations based on simple models

H9 develops and carries out statistical processes to answer questions which she/he and others have posed

H11 uses mathematical argument and reasoning to evaluate conclusions drawn from other sources, communicating his/her position clearly to others.


identifying measures of location as mean and median

identifying measures of spread as range, interquartile range and standard deviation

investigating outliers in small data sets and their effects on the mean, median and mode

describing the general shape of a graph or display which represents a given data set, eg in terms of smoothness, symmetry or number of modes

making judgements about the data based on observed features of the display such as shape and skewness

displaying data in double (back-to-back) stem-and-leaf plots

displaying data in two box-and-whisker plots drawn on the same scale

displaying two sets of data on a radar chart

preparing an area chart to illustrate and compare different sets of data over time (see example at end of unit)

using multiple displays to describe and interpret the relationships between data sets

interpreting data presented in two-way table form, eg male/female versus exercise/no exercise

comparing summary statistics from two sets of data.


area chart outlier skewness.Technology that may be used in support of this unit

57


use of a spreadsheet to create frequency tables and calculate mean and standard deviation

calculate mean and standard deviation through the use of inbuilt statistical functions of a calculator.

Suggested applications and modelling tasks (DA5) Prepare a stem-and-leaf plot from collected data. Use it to decide whether clustering is

present, whether the shape of the display indicates any skewness of scores or if there is any other tendency in the data.

Students could use summary statistics and statistical displays to compare related pairs of data sets to decide, informally, whether they could have been drawn from the same population or whether they appear to be from different populations. The following (indicative) pairs of data sets could be considered:– home scores versus away scores in national sporting competitions – ages of Oscar-winning male actors versus ages of Oscar-winning female actors – blood pressure of males versus that of females– heights of male versus heights of female students (restricted age range)– customer waiting times on two different days of the week at a fast-food outlet– monthly rainfall in mm for different cities or regions, eg on average does it rain more

in Melbourne than in Sydney? Is it a significant difference? Students could discuss what is meant by ‘significant’ in this context.

– population by age bands for different countries or regions. Investigate population pyramid histograms. Given data on a particular population in a table such as the one shown below, answer

questions like ‘What percentage of the women surveyed in this exercise are smokers?’ ‘What percentage of smokers surveyed in this exercise are women?’ Discuss the difference between these questions. Under what circumstances would the percentages be the same?

men women TOTALSsmokers 4027 4426 8453

non-smokers 8321 7462 15 783TOTALS 12 348 11 888 24 236

58


DA6: The normal distribution

In this unit, students will apply the properties of the standard normal distribution to the solution of real problems.

Outcomes addressed

A student:



H5 makes predictions about the behaviour of situations based on simple modelsH9 develops and carries out statistical processes to answer questions which she/he and

others have posedH11 uses mathematical argument and reasoning to evaluate conclusions drawn from other



describing the z-score (standardised score) corresponding to a particular score in a set of scores as a number indicating the position of that score relative to the mean

using the formula to calculate z-scores, where s is the standard deviation

( for a population, for a sample)

using calculated z-scores to compare scores from different data sets identifying the properties of data that are normally distributed, ie

– the mean, median and mode are equal – if represented by a histogram, the resulting frequency graph is ‘bell shaped’

using collected data to illustrate that, for normally distributed data:– approximately 68% of scores will have z-scores between –1 and 1– approximately 95% of scores will have z-scores between –2 and 2– approximately 99.7% of scores will have z-scores between –3 and 3

using these measures to make judgements in individual cases.


normal distributionstandardised scorez-score.

59



Determine the proportion of individuals in a normal population that lies within one standard deviation of the mean.

Packets of rice are each labelled as having a mass of 1 kg. The mass of these packets is normally distributed with a mean of 1.02 kg and a standard deviation of 0.01kg. Complete the following table:

Mass in kg 1.00 1.01 1.02 1.03 1.04

z-score 0 1

(a) What percentage of packets will have a mass less than 1.02 kg?(b) What percentage of packets will have a mass between 1.00 and 1.04 kg?(c) What percentage of packets will have a mass between 1.00 and 1.02 kg?(d) What percentage of packets will have a mass less than the labelled mass?

A machine is set for the production of cylinders of mean diameter 5.00 cm with standard deviation 0.020 cm. Within what intervals will 99% of the diameters lie?

If a cylinder, randomly selected from this production, has a diameter of 5.070 cm, what conclusion could be drawn?

Given the frequency table of a familiar characteristic, students should analyse the data to determine whether or not it could be considered normally distributed.

Students should investigate whether the results of a particular experiment would be normally distributed.

Compare student performances in different tests to establish which is the ‘better’ performance.

60


DA7: Correlation

In this unit, students investigate the strength of association of data through examining a scatterplot of ordered pairs. Where appropriate, students find the equation of a line of fit and use the equation to make predictions.

Outcomes addressed

A student:




H5 makes predictions about the behaviour of situations based on simple modelsH9 develops and carries out statistical processes to answer questions which she/he and

others have posedH11 uses mathematical argument and reasoning to evaluate conclusions drawn from other



plotting ordered pairs of data onto a scatterplot recognising from the scatterplot:

1. whether the points appear to form a mathematical pattern2. whether the pattern appears to be linear

establishing a median regression line to give a line of fit on a scatterplot with a ruler and pencil

measuring the gradient of the line of fit drawn, with ruler and pencil noting the vertical intercept of the line of fit drawn establishing the equation of the resulting line of fit in form (see AM2) using this equation to make predictions.

The remaining points relate to correlation. Students will not be required to calculate correlation coefficients.

interpreting the strength of association using a given correlation coefficient interpreting the sign of a given correlation coefficient recognising that a high degree of correlation does not necessarily imply causality, eg

there is a very high correlation between the sizes of one’s left and right feet, but one does not cause the other.

61


Terminology introduced in this unitcausalitycorrelationcorrelation coefficient line of best fitmedian regression linescatterplot.


use a graphing facility or graphing calculator to create a scatter plot from two sets of data and hence determine the equation of the line of best fit.


Students could collect data and model the following situations: (Consideration could include the strength of association and whether either variable can be used as a predictor of the other.)– hand span versus height– height versus weight– world sporting records versus time– time to run 400 metres versus resting heart rate– time to run 400 metres versus height– height versus age of children (aged from 0 to 12 years)– barometer readings versus maximum daily temperature– Mathematics test scores versus Science test scores (or English test scores).

Plot a scatterplot from experimental data and interpret the diagram in terms of the theoretical situation.

Evaluate reports and media articles in which information has been collected which suggests relationships between two sets of data, eg smoking and lung cancer.

Discuss relationships between variables and whether these imply cause and effect, eg height and weight, the number of hotels and churches in rural towns, smoking and lung cancer, homework hours and test marks.

62


9.3 Measurement

M5: Further applications of area and volume

In this unit, the work commenced in M2: Applications of area and volume is extended to include surface area and volume of complex figures, and the use of approximations in calculating area and volume of irregular figures.

Outcomes addressed

A student:



H3 develops and tests a general mathematical relationship from observed patternsH6 analyses two-dimensional and three-dimensional models to solve practical and

mathematical problemsH7 interprets the results of measurements and calculations and makes judgements about

reasonablenessH11 uses mathematical argument and reasoning to evaluate conclusions drawn from other



calculating areas of ellipses, annuluses and parts of a circle (quadrant, sector) using appropriate formulae

calculating areas of composite figures applying Simpson’s rule over three equally spaced points, ie one application (problems

involving five points should be treated using two applications ) calculating external surface area of open (without top and/or bottom) and closed

cylinders calculating surface area of spheres calculating volumes of composite solids determining errors in calculations resulting from errors in measurement.


annulusellipsequadrantsector.

63



use of drawing tools to create 2D and 3D drawings.


Design the shape and dimensions of a container that would have a given capacity, given the purpose and use of the container.

Design cost-effective packaging. For example, groups of students are given four table-tennis balls and need to design a box to package them which minimises the use of material.

As designers, students are told that they have been given a square piece of metal of side length 2 m from which to design an open rectangular water tank. The volume of water that the tank will hold will depend on the size of squares cut from each of the four corners of the square. Students choose a scale and make models of tanks and find the volume of water that they hold. They could graph results and determine when the volume is the greatest. What happens if the side of the original square is doubled?

Students use Simpson’s rule to estimate the volume of a dam by calculating a series of equally spaced cross-sectional areas.

Students calculate the cross-sectional area of a river, and hence calculate the flow-rate of the river.

64


M6: Applications of trigonometry

This unit extends students’ knowledge of trigonometry and area to include non-right- angled triangles. Problems to be solved will incorporate practical work with offset and radial surveys. Angles will be approximated to the nearest minute.

Outcomes addressed

A student:



H6 analyses two-dimensional and three-dimensional models to solve practical and mathematical problems

H7 interprets the results of measurements and calculations and makes judgements about reasonableness



solving problems using trigonometric ratios in one or more right-angled triangles using compass bearings (eight points only) and true bearings (three-figure bearings) in

problem-solving related to maps and charts establishing the sine, cosine and tangent ratios for obtuse angles from a calculator determining the sign of the above ratios for obtuse angles preparing diagrams to represent given information using the sine rule to find lengths and angles

Note: It is not intended that students study the ambiguous case of the sine rule. calculating area of a triangle using the formula

using the cosine rule to find lengths and angles

or

65


using appropriate trigonometric ratios and formulae in two-triangle problems where one triangle is right-angled and the diagram is given

solving problems involving non-right-angled triangles selecting and using appropriate trigonometric ratios and formulae to solve problems conducting radial (both plane table and compass) surveys solving problems involving non-right-angled triangle trigonometry, Pythagoras’ theorem

and area in offset and radial surveys.


compass radial surveycosine ruleoffset surveyplane table radial surveysine rule. Suggested applications and modelling tasks (M6)

Students estimate the area and perimeter of a part of the school grounds. They carry out a survey and complete a scale diagram, which they use to calculate the area and perimeter. Students present, compare and discuss their results. Would a different type of survey give different results?

Use a large area, wool and ‘Blu-tac’ to demonstrate how to draw diagrams resulting from problems involving bearings. Compass points, angles and distances can be marked with chalk. The diagram is created with wool, which is anchored by ‘Blu-tac’ or plasticine. As each part of the problem is read and clarified, students discuss what to do and the wool is extended to the next position, where it is anchored.

Carry out a radial survey of an irregular area and compare the result with an offset survey.

Plan a walk or some other journey by reading and interpreting a map. Find gradients from contour lines on maps. Use navigational charts to plan routes and identify positions. Calculate the height of buildings and other structures. Investigate the derivation of the formulae used in this section.

66


M7: Spherical geometry

In this unit, geometry and trigonometry are applied to solve problems relating to the Earth as a sphere. Applications include locating positions on the Earth, using latitude and longitude, and calculating time differences.

Outcomes addressed

A student:



H6 analyses two-dimensional and three-dimensional models to solve practical and mathematical problems

H7 interprets the results of measurements and calculations and makes judgements about reasonableness



calculating arc lengths of a circle distinguishing between great and small circles using the Equator and the Greenwich Meridian as lines of reference for locations on the

Earth’s surface locating positions on the globe using latitude and longitude converting nautical miles (M) to kilometres and vice versa, given 1.852 km = 1 M calculating distances between two points on the same great circle in nautical miles and

kilometres (radius of the Earth to be taken as 6400 km) defining 1 knot as a speed of 1 M per hour using time zones and the International Date Line in solving problems calculating time differences given the difference in longitudes

(Apply 15° = 1 hour and 1°= 4 minutes time difference. Daylight-saving time is to be considered.)

determining times for cities in different countries in related travel questions.


great circleInternational Date Linelatitudemeridian of longitudeparallel of latitude

Greenwich Meridian knot longitude nautical milesmall circle.

67



Calculate the present time in all the capital cities of Australia. Investigate which States in Australia have daylight-saving, and if so, when it starts and

finishes. Students collect data to enable them to discuss and present information on the

popularity of daylight-saving. Students find three cities with the same time, say a given number of hours earlier and

later than local time. These could be marked on a world map and displayed to reinforce the concept of time zones.

Find the coordinates of the point on the Earth’s surface that is at maximum distance from your current location.

68


9.4 Probability

PB3: Multi-stage events

The focus of this unit is on counting the number of outcomes for an experiment, or the number of ways in which an event may occur. The probability of particular outcomes may then be established. The formulae using factorial notation are not required.

Outcomes addressed

A student:


H3 develops and tests a general mathematical relationship from observed patternsH4 analyses representations of data in order to make inferences, predictions and

conclusionsH10 solves problems involving uncertainty using basic principles of probabilityH11 uses mathematical argument and reasoning to evaluate conclusions drawn from other



constructing and using a tree diagram to establish the sample space for a simple multi-stage event

multiplying the number of choices at each stage to determine the number of outcomes for a multi-stage event

establishing that the number of ways in which n different items can be arranged is, eg the number of arrangements of 4 different items is

; the number of arrangements of 3 different items is

checking that these results are true by listing arrangements for small numbers of items establishing the number of ordered selections that can be made from a group of

different items (small numbers only), eg if selecting two particular positions (such as captain and vice-captain) from a team of five people, the number of selections is

verifying these results by listing ordered selections (small numbers only) establishing the number of unordered selections that can be made from a group of

different items (small numbers only), eg if selecting a pair of people to represent a team of five, the number of selections is half of the number of ordered selections

verifying by listing unordered selections (small numbers only) using the formula for the probability of an event to calculate the probability that a

particular selection will occur using probability tree diagrams to solve problems involving two-stage events.

69



ordered selection probability tree diagram tree diagram two stage events unordered selection.


Determine the total number of choices in a game in which six different numbers are chosen from 40.

Use Venn diagrams to describe and interpret situations with intersecting sample spaces.

Choosing cards from a pack of cards, with and without replacement, could be used to investigate two-stage experiments.

Select a ‘triple-decker’ (three flavours) ice-cream from four different flavours: chocolate, strawberry, vanilla and butterscotch. If you are fussy about the order in which you eat the scoops, your choices are ordered. Show, with a list and a tree diagram, that there are 24 possible arrangements for your ice-cream. What is the probability that the chocolate scoop will be on the top? If you are happy to put all three scoops together in a tub and eat them all at once, how many different choices do you have?

How many different ways are there of answering a four question True/False test? Check by listing the possible responses.

In how many ways can the names of three candidates be listed on a ballot paper? If you are one of the candidates, what is the probability that your name will be at the top of the paper? Check by listing.

A road toll of $1.50 is paid in coins at a toll gate. List the different ways in which you could pay this toll, assuming that any combination of $1, 50c, 20c and 10c coins may be used.

A new model car is available with leatherette or cloth upholstery, and four exterior paint colours. There is also an option for air-conditioning, and an option for a CD player. If a car dealer wishes to display one car for each possible combination, how many cars would be needed?

70


PB4: Applications of probability

In this unit, students calculate expected outcomes from simple experiments and compare them with experimental results.

Outcomes addressed

A student:



H10 solves problems involving uncertainty using basic principles of probabilityH11 uses mathematical argument and reasoning to evaluate conclusions drawn from other



calculating the expected number of times a particular outcome would arise, given the number of trials of a simple experiment, by establishing the theoretical probability and multiplying by the number of trials

comparing the above result with an experimental result calculating financial expectation by multiplying each financial outcome by its probability

and adding the results together Note: A financial loss is regarded as negative.

carrying out simulations to model events, eg tossing a coin with the outcomes representing the sex of the offspring

drawing up a table (two-way table) to illustrate results gained on a test designed to determine the existence (in a particular case) of a phenomenon which has a low overall probability of occurrence, eg screening for medical conditions, or using a lie detector to indicate guilt or innocence

interpreting the information in the table and making judgements about the conclusions established by the test.


expected outcomefinancial expectationsimulationtwo-way table.

71



using a spreadsheet to simulate large numbers of trials using inbuilt graphing facilities of the spreadsheet to investigate the results of a

simulation.


Determine whether it is better to buy 10 tickets in one lottery, or one ticket in each of 10 lotteries.

Given the following options, most people would prefer (A) sure gain of $300

(B) 30% chance of gaining $1000 and 70% chance of gaining nothing Use expectation to decide whether this preference is justified. Investigate different strategies for playing party games such as ‘Greedy Pig’. Simulate the number of boys and girls in a large number of families to answer questions

about the most likely distributions under different circumstances, eg if couples stopped having children as soon as the first girl was born, would there be more girls than boys in the long run?

Paul plays a game involving the tossing of two coins. He gains $5 if they both show heads and $1 for a head and a tail, but loses $6 if they both show tails. What is his financial expectation from this game?

72


9.5 Algebraic Modelling

AM3: Algebraic skills and techniques

This unit develops algebraic skills and techniques that are used in work-related and everyday contexts. As far as possible, real contexts should be used to demonstrate the use of algebra in practical life.

Outcomes addressed

A student:


H3 develops and tests a general mathematical relationship from observed patternsH7 interprets the results of measurements and calculations and makes judgements about

reasonablenessH11 uses mathematical argument and reasoning to evaluate conclusions drawn from other



substituting into and evaluating algebraic expressions — linear, quadratic,

cubic, as well as those involving square and cube roots, such as r =3V4π

3

adding and subtracting like terms multiplying and dividing algebraic terms and expressions changing the subject of equations and formulae involving linear and quadratic terms, eg

make s the subject of v2 =u2 + 2as

solving equations after substituting values, eg evaluate t when d =5t2 and d = 300

solution of equations arising from practical situations by estimation and refinement, eg if x is the number of years required for an investment to double at 5% pa compound interest, then 1.05x =2 . Find the value of x by making an initial estimate and refining the result, using a calculator

using positive and negative powers of ten as part of expressing measurements in scientific notation.

73



Washing removes 20% of a deep stain at each wash. Show that three washes remove about half of the stain. How many washes are needed to reduce a stain to 5% of its original amount?

My photocopier enlarges by 150%. Can I produce a picture about twice the size of an original? Explain how this could be done.

The distance that an object falls (d metres) in a certain time (t seconds) is approximated by d =5t2 . Create a table of values for this function manually, and compare this result with that obtained through the use of a spreadsheet. How long does it take for an object to fall 300 m?

74


AM4: Modelling linear and non-linear relationships

This unit focuses on the examination of practical problems that can be modelled algebraically.

Outcomes addressed

A student:



H3 develops and tests a general mathematical relationship from observed patternsH5 makes predictions about the behaviour of situations based on simple modelsH11 uses mathematical argument and reasoning to evaluate conclusions drawn from other



generating tables of values and graphing linear functions with pencil and paper interpretation of the point of intersection of the graphs of two linear functions drawn

from practical contexts, eg ‘breakeven’ points (see applications) generating tables of values and graphing quadratic functions of the form

y=ax2 +bx+ c, x≥0 with pencil and paper

noting that different forms of an expression produce identical graphs, eg y= x−2( )2 + 3, y=x2 −4x+ 7

using a quadratic graph to find maximum and minimum values in practical contexts generating tables of values and graphing cubic, exponential and hyperbolic functions

with pencil and paper. Functions to be restricted to the following forms – cubic:

y=ax3, x≥0 ; exponential: y=b ax( ), x≥0 ; hyperbolic y=

ax, x> 0

recognition that, for a > 1, y = b(ax) represents exponential growth and, for 0 < a < 1, it represents exponential decay (see FM2, FM6)

development of equations such as y = ax2, h = at3 from descriptions of situations in which one quantity varies directly as the power of another

development of an equation such as y=ax from a description of a situation in

which one quantity varies inversely with another subsequent evaluation of a in the equations shown in the above two points, given one

pair of variables, and using the resulting formula to find other values of the variables using algebraic functions as models of physical phenomena recognising the limitations of models when interpolating and/or extrapolating.

75



exponential functionextrapolate/interpolate hyperbolic functionquadratic functions.


use graphing software (or a graphing calculator) to: graph linear, quadratic, cubic and exponential functions find graphically a point of intersection using ‘zooming’ techniques identify ordered pairs on a graph by ‘tracing’ determine the equation of a line of best fit on a scatterplot.


Distance/velocity/time graphs and problems. Is the approximation method ‘Double and add 30˚ for converting from Celsius to

Fahrenheit always close? The formula for converting degrees Celsius to degrees Fahrenheit could be graphed, along with the formula arising from the ‘rule of thumb’. This could also be investigated with a spreadsheet and/or a graphing calculator.

Create graphs to demonstrate the solution of simultaneous linear equations is the location of ‘breakeven’ points, eg in graphs of costs and income, or in finding the time when a newly installed appliance starts to save money for a household.

Sketch at least ten rectangles that have the same perimeter. Record length versus area in a table. Plot the resulting function and use the graph to determine the shape with maximum area.

Investigate models of population growth. Plot a function showing the growth of a population of bacteria. Investigate compound interest as the time period shortens. Investigate via technology the effect of adding/subtracting a constant term on the graph

of a function.

On the Earth, the formula d = 5t2 can be used to express the distance (d metres) that an object falls in t seconds, if air resistance can be ignored. Investigate equivalent expressions for other bodies, eg on the moon, the equation is d = 0.8t2.

Solve problems where two quantities vary in a related way. For example, the number of eggs used in a recipe for a particular cake varies with the square of the diameter of the tin, for tins with constant depth. If two eggs are used in a recipe for a 15 cm diameter tin, how many eggs would be used if the diameter of the tin is 35 cm?

76


The surface area of a sphere can be approximated by S = 3D, where D is thediameter. Compare results from a number of spheres with those obtained using S =πD2 . (Note: S and D are in the same basic unit, eg if D is in cm, S will be in cm2).

Investigate reaction times and stopping distances for cars travelling at different speeds. For example, a car travelling at v km/h will travel 0.7v m during the 2.5 seconds it takes the driver to react to a dangerous situation. For a car with excellent brakes on a dry level road, the total distance travelled by a car once the driver begins to realize that there is a need to stop can be expressed by d =0.01v2 + 0.7v , if there is no skidding .

Cubic models can be used to estimate the mass of objects. For example, could you lift a cubic metre of cork? An expression for the mass in grams, M, of a cube of cork is M =0.25x3 , where x is the side length of the cube in centimetres.

Hyperbolic models can be used to find how much each person contributes when a cost is shared. For example, a household has $306 in bills. Make a table and a graph to show how much each person pays if there are 2, 3, 4 or 5 people contributing equally to pay these bills.

The time in seconds taken to travel 100 m is inversely related to the speed in kilometres per hour. Students could illustrate this with a table, a graph and a formula.

An exponential expression such as M =1.5 1.2( )x can be used to describe the mass M kg of a baby orang-utang at age x months. This model applies for a limited time, up to x = 6.

The size of the world’s population in billions can be approximated by the expression y=2.486 1.02( )x , where x is the number of years since 1950 and 2.486 billion was the size of the population in 1950. This model could be checked against recent data. Similar expressions can be developed in different countries and cities. If a city has a declining population, for example a decline of 5% pa from an initial population of 100 000, the equation is y=100 000 0.95( )x .

Review models of depreciation, inflation and appreciation.

Graph and compare the functions y=1000 1.05( )x and y=1000 0.95( )x . What might these graphs represent?

77


10 Course Requirements

The General Mathematics Stage 6 Syllabus includes a Preliminary course of 120 hours (indicative time) and an HSC course of 120 hours (indicative time).

The General Mathematics Preliminary course is constructed on the assumption that students have achieved the outcomes in the core of the Standard Mathematics course for the School Certificate, together with the recommended options: Trigonometry and Further Algebra. Completion of the Preliminary course is a prerequisite for the study of the HSC course.

Students may not study General Mathematics in conjunction with any other Mathematics course in Stage 6.

78


11 Post-school Opportunities

The study of General Mathematics Stage 6 provides students with knowledge, understanding and skills that form a valuable foundation for a range of courses at university and other tertiary institutions.

In addition, the study of General Mathematics Stage 6 assists students to prepare for employment and full and active participation as citizens. In particular, there are opportunities for students to gain recognition in vocational education and training. Teachers and students should be aware of these opportunities.

Recognition of Student Achievement in Vocational Education and Training (VET)

Wherever appropriate, the skills and knowledge acquired by students in their study of HSC courses should be recognised by industry and training organisations. Recognition of student achievement means that students who have satisfactorily completed HSC courses will not be required to repeat their learning in courses in TAFE NSW or other Registered Training Organisations (RTOs).

Registered Training Organisations, such as TAFE NSW, provide industry training and issue qualifications within the Australian Qualifications Framework (AQF).

The degree of recognition available to students in each subject is based on the similarity of outcomes between HSC courses and industry training packages endorsed within the AQF. Training packages are documents that link an industry’s competency standards to AQF qualifications. More information about industry training packages can be found on the National Training Information Service (NTIS) website (www.ntis.gov.au).

Recognition by TAFE NSW

TAFE NSW conducts courses in a wide range of industry areas, as outlined each year in the TAFE NSW Handbook. Under current arrangements, the recognition available to students of mathematics in relevant courses conducted by TAFE is described in the HSC/TAFE Credit Transfer Guide. This guide is produced by the Board of Studies and TAFE NSW and is distributed annually to all schools and colleges. Teachers should refer to this guide and be aware of the recognition that may be available to their students through the study of General Mathematics Stage 6. This information can be found on the TAFE NSW website (www.tafensw.edu.au/mchoice).

Recognition by other Registered Training Organisations

Students may also negotiate recognition into a training package qualification with another Registered Training Organisation. Each student will need to provide the RTO with evidence of satisfactory achievement in General Mathematics Stage 6 so that the degree of recognition available can be determined.

79


12 Assessment and Reporting

Advice on appropriate assessment practice in relation to the General Mathematics course is contained in the document Assessment and Reporting in Stage 6 General Mathematics.

This document provides general advice on assessment in Stage 6 as well as the specific requirements for the Preliminary and HSC courses. The document contains:

suggested components and weighting for the internal assessment of the Preliminary coursemandatory components and weightings for the internal assessment of the HSC coursethe HSC examination specifications, which describe the format of the external HSC examination.

This document and other resources and advice related to assessment in Stage 6 General Mathematics are available on the Board's website at www.boardofstudies.nsw.edu.au/syllabus_hsc

80

http://www.boardofstudies.nsw.edu.au/syllabus_hsc

General Maths HSC Syllabus

Documents

Transcript of General Maths HSC Syllabus